Overview

rank(A) = the maximum number of linearly independent rows or columns of A.

If one considers the matrix as a linear transformation from Rn to Rm, then the column space of the matrix equals the image of this linear transformation.

The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a1, ...., an], then colsp(A) = span {a1, ...., an}.

The concept of row space generalizes to matrices over C, the field of complex numbers, or over any field.

Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it will (1) first project x into the row space of A, (2) perform an invertible transformation, and (3) place the resulting vector y in the column space of A. Thus the result y = Ax must reside in the column space of A. See singular value decomposition for more details on this second interpretation.

Example

Given a matrix J:

the rows are
r1 = (2,4,1,3,2),
r2 = (−1,−2,1,0,5),
r3 = (1,6,2,2,2),
r4 = (3,6,2,5,1).
Consequently, the row space of J is the subspace of R5spanned by { r1, r2, r3, r4 }.
Since these four row vectors are linearly independent, the row space is 4-dimensional. Moreover, in this case it can be seen that they are all orthogonal to the vector n = (6,−1,4,−4,0), so it can be deduced that the row space consists of all vectors in R5 that are orthogonal to n.

Column space

Definition

Let K be a field of scalars. Let A be an m×n matrix, with column vectors v1,v2,...,vn. A linear combination of these vectors is any vector of the form

where c1,c2,...,cn are scalars. The set of all possible linear combinations of v1,...,vn is called the column space of A. That is, the column space of A is the span of the vectors v1,...,vn.

Any linear combination of the column vectors of a matrix A can be written as the product of A with a column vector:

Therefore, the column space of A consists of all possible products Ax, for x∈Cn. This is the same as the image (or range) of the corresponding matrix transformation.

Example

If , then the column vectors are v1=(1,0,2)T and v2=(0,1,0)T.

A linear combination of v1 and v2 is any vector of the form

The set of all such vectors is the column space of A. In this case, the column space is precisely the set of vectors (x,y,z)∈R3 satisfying the equation z=2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space).

At this point, it is clear that the first, second, and fourth columns are linearly independent, while the third column is a linear combination of the first two. (Specifically, v3=–2v1+v2.) Therefore, the first, second, and fourth columns of the original matrix are a basis for the column space:

Note that the independent columns of the reduced row echelon form are precisely the columns with pivots. This makes it possible to determine which columns are linearly independent by reducing only to echelon form.

The above algorithm can be used in general to find the dependence relations between any set of vectors, and to pick out a basis from any spanning set. A different algorithm for finding a basis from a spanning set is given in the row space article; finding a basis for the column space of A is equivalent to finding a basis for the row space of the transpose matrixAT.

Dimension

The dimension of the column space is called the rank of the matrix. The rank is equal to the number of pivots in the reduced row echelon form, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4×4 matrix in the example above has rank three.

Because the column space is the image of the corresponding matrix transformation, the rank of a matrix is the same as the dimension of the image. For example, the transformation R4→R4 described by the matrix above maps all of R4 to some three-dimensional subspace.

The nullity of a matrix is the dimension of the null space, and is equal to the number of columns in the reduced row echelon form that do not have pivots.[3] The rank and nullity of a matrix A with n columns are related by the equation:

Relation to the left null space

The left null space of A is the set of all vectors x such that xTA=0T. It is the same as the null space of the transpose of A. The product of the matrix AT and the vector x can be written in terms of the dot product of vectors:

because row vectors of AT are transposes of column vectors vk of A. Thus ATx=0 if and only if x is orthogonal (perpendicular) to each of the column vectors of A.

It follows that the left null space (the null space of AT) is the orthogonal complement to the column space of A.

For a matrix A, the column space, row space, null space, and left null space are sometimes referred to as the four fundamental subspaces.

For matrices over a ring

Similarly the column space (sometimes disambiguated as right column space) can be defined for matrices over a ringK as

for any c1,...,cn, with replacement of the vector m-space with "rightfree module", which changes the order of scalar multiplication of the vector vk to the scalar ck such that it is written in an unusual order vector–scalar.[4]

Row space

Definition

where c1,c2,...,cm are scalars. The set of all possible linear combinations of r1,...,rm is called the row space of A. That is, the row space of A is the span of the vectors r1,...,rm.

For example, if

then the row vectors are r1=(1,0,2) and r2=(0,1,0). A linear combination of r1 and r2 is any vector of the form

The set of all such vectors is the row space of A. In this case, the row space is precisely the set of vectors (x,y,z)∈K3 satisfying the equation z=2x (using Cartesian coordinates, this set is a plane through the origin in three-dimensional space).

For a matrix that represents a homogeneous system of linear equations, the row space consists of all linear equations that follow from those in the system.

Once the matrix is in echelon form, the nonzero rows are a basis for the row space. In this case, the basis is {(1,3,2),(0,1,0)}. Another possible basis {(1,0,2),(0,1,0)} comes from a further reduction.[5]

This algorithm can be used in general to find a basis for the span of a set of vectors. If the matrix is further simplified to reduced row echelon form, then the resulting basis is uniquely determined by the row space.

It is sometimes convenient to find a basis for the row space from among the rows of the original matrix instead (for example, this result is useful in giving an elementary proof that the determinantal rank of a matrix is equal to its rank). Since row operations can affect linear dependence relations of the row vectors, such a basis is instead found indirectly using the fact that the column space of AT is equal to the row space of A. Using the example matrix A above, find AT and reduce it to row echelon form:

The pivots indicate that the first two columns of AT form a basis of the column space of AT. Therefore, the first two rows of A (before any row reductions) also form a basis of the row space of A.

Dimension

The dimension of the row space is called the rank of the matrix. This is the same as the maximum number of linearly independent rows that can be chosen from the matrix, or equivalently the number of pivots. For example, the 3×3 matrix in the example above has rank two.[5]

The rank of a matrix is also equal to the dimension of the column space. The dimension of the null space is called the nullity of the matrix, and is related to the rank by the following equation:

where n is the number of columns of the matrix A. The equation above is known as the rank–nullity theorem.

Relation to the null space

The null space of matrix A is the set of all vectors x for which Ax=0. The product of the matrix A and the vector x can be written in terms of the dot product of vectors:

where r1,...,rm are the row vectors of A. Thus Ax=0 if and only if x is orthogonal (perpendicular) to each of the row vectors of A.

It follows that the null space of A is the orthogonal complement to the row space. For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin. This provides a proof of the rank–nullity theorem (see dimension above).

Relation to coimage

If V and W are vector spaces, then the kernel of a linear transformationT:V→W is the set of vectors v∈V for which T(v)=0. The kernel of a linear transformation is analogous to the null space of a matrix.

If V is an inner product space, then the orthogonal complement to the kernel can be thought of as a generalization of the row space. This is sometimes called the coimage of T. The transformation T is one-to-one on its coimage, and the coimage maps isomorphically onto the image of T.

When V is not an inner product space, the coimage of T can be defined as the quotient spaceV/ker(T).

See also

Notes

↑ Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Lay 2005, Meyer 2001, and Strang 2005.